Root Locus Method Example, PID Controller

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Root Locus Method Example, PID Controller

• Modern Control Systems
• Lecture 17

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Outline

• Example to illustrate root locus procedure
• PID controller
• The root locus using Matlab

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Seven steps to sketching a root locus
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Example a 4th-order system to illustrate the
root locus procedure

• Step 1. Prepare the root locus sketch.
• (a) We desire to plot the root locus for the
  characteristic equation of a system as K varies
  for Kgt0 when

(b) Factor P(s) in terms of n poles and M zeros.
(c) Locate the open-loop poles and zeros in
s-plane. The system has 4 poles and no finite
zeros.
(d) Determine the number of separate loci. We
have 4.
(e) The root loci are symmetrical with respect to
the real axis.
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• Step 2. Locate the segments of the real axis that
  are root loci.
• The root locus on the real axis always lies in a
  section of the real axis to the left of an odd
  number of poles.

A segment of the root locus exists on the real
axis between s0 and s-4.
Step 3. The loci proceed to the zeros at 8 along
asymptotes with angles
and centred at
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• Step 4. Determine the points at which the locus
  crosses j?-axis if it does so. We need to use
  Routh-Hurwitz criterion.

The characteristic equation is rewritten as
The Routh array is
Therefore, the limiting value of K for stability
is
and when K568.89, the auxiliary polynomial is
Hence, when K568.89, the root locus crosses
j?-axis at
7

• Step 5. Determine the breakaway points on the
  real axis if any.

Thus, the breakaway point on the real axis is at
s-1.5767.
Step 6. Determine departure and/or arrival angles
using the angle condition.
root locus
Step 7. Complete the root locus sketch.
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PID controllers

• One form of controller widely used in industrial
  process control is called PID controller. It
  provides a proportional term, an integration term
  and a derivative term. The TF of the PID
  controller is

If we set KD0, we have the PI controller
If we set KI0, we have the PD controller
To implement a PID controller, three parameters
must be determined, KP, KI and KD.
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• Consider the PID controller

Therefore, the PID controller introduces a TF
with one pole at the origin and two zeros that
can be located anywhere in the left hand s-plane.
Consider the closed-loop system in the figure,
where
We use a PID controller with complex zeros,
z1-3j1 and z2z1. We can plot the root locus.
root locus
From the root locus, as KD increases, the
closed-loop roots approach z1 and z2. Hence, we
can select z1 and z2 and set KD to adjust the
transient response of the system.
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Root locus using Matlab

• An approximate root locus sketch can be obtained
  by applying the orderly procedure we have
  learned. Matlab can provide an accurate root
  locus plot. But we should not rely solely on
  Matlab without understanding the root locus
  concepts that are embedded in the manual steps.
• The Matlab functions for obtaining root locus
  plots are rlocus, rlocfind.

To use the rlocus function for generating root
locus plots, the characteristic equation should
be written in root locus form
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Root locus using Matlab (contd)
To find the value of K corresponding to certain
roots, we can use the rlocfind function, but only
after the root locus has been generated by the
rlocus function.